数学专业英语2.docx
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数学专业英语2.docx
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数学专业英语2
MathematicalEnglish
Dr.XiaominZhang
Email:
zhangxiaomin@
§2.2GeometryandTrigonometry
TEXTAWhystudygeometry?
Whydowestudygeometry?
Thestudentbeginningthestudyofthistextmaywellask,“Whatisgeometry?
WhatcanIexpecttogainfromthisstudy?
”
Manyleadinginstitutionsofhigherlearninghaverecognizedthatpositivebenefitscanbegainedbyallwhostudythisbranchofmathematics.Thisisevidentfromthefactthattheyrequirestudyofgeometryasaprerequisitetomatriculationinthoseschools.
GeometryhaditsoriginlongagointhemeasurementsbytheBabyloniansandEgyptiansoftheirlandsinundatedbythefloodsoftheNileRiver.TheGreekwordgeometryisderivedfromgeo,meaning“earth”,andmetron,meaning“measure”.Asearlyas2000B.C.wefindthelandsurveyorsofthesepeoplere-establishingvanishinglandmarksandboundariesbyutilizingthetruthsofgeometry.
Geometryisasciencethatdealswithformsmadebylines.Astudyofgeometryisanessentialpartofthetrainingofthesuccessfulengineer,scientist,architect,anddraftsman.Thecarpenter,machinist,stonecutter,artist,anddesignerallapplythefactsofgeometryintheirtrades.Inthiscoursethestudentwilllearnagreatdealaboutgeometricfiguressuchaslines,angles,triangles,circles,anddesignsandpatternsofmanykinds.
Oneofthemostimportantobjectivesderivedfromastudyofgeometryismakingthestudentbemorecriticalinhislistening,reading,andthinking.Instudyinggeometryheisledawayfromthepracticeofblindacceptanceofstatementsandideasandistaughttothinkclearlyandcriticallybeforeformingconclusions.
Therearemanyotherlessdirectbenefitsthestudentofgeometrymaygain.AmongtheseonemustincludetrainingintheexactuseoftheEnglishlanguageandintheabilitytoanalyzeanewsituationorproblemintoitsbasicparts,andutilizingperseverance,originality,andlogicalreasoninginsolvingtheproblem.Anappreciationfortheorderlinessandbeautyofgeometricformsthataboundinman’sworksandofthecreationsofnaturewillbeabyproductofthestudyofgeometry.Thestudentshouldalsodevelopanawarenessofthecontributionsofmathematicsandmathematicianstoourcultureandcivilization.
TEXTBSomegeometricalterms
1.Solidsandplanes.Asolidisathree-dimensionalfigure.Commonexamplesofsolidarecube,sphere,cylinder,coneandpyramid.
Acubehassixfaceswhicharesmoothandflat.Thesefacesarecalledplanesurfacesorsimplyplanes.Aplanesurfacehastwodimensions,lengthandwidth.Thesurfaceofablackboardoratabletopisanexampleofaplanesurface.
2.Linesandlinesegments.Weareallfamiliarwithlines,butitisdifficulttodefinetheterm.Alinemayberepresentedbythemarkmadebymovingapencilorpenacrossapieceofpaper.Alinemaybeconsideredashavingonlyonedimension,length.Althoughwhenwedrawalinewegiveitbreadthandthickness,wethinkonlyofthelengthofthetracewhenconsideringtheline.Apointhasnolength,nowidth,andnothickness,butmarksaposition.Wearefamiliarwithsuchexpressionsaspencilpointandneedlepoint.Werepresentapointbyasmalldotandnameitbyacapitalletterprintedbesideit,as“pointA”inFig.2-2-1.
Thelineisnamedbylabelingtwopointsonitwithcapitallettersoronesmallletternearit.ThestraightlineinFig.2-2-2isread“lineAB”or“linel”.Astraightlineextendsinfinitelyfarintwodirectionsandhasnoends.Thepartofthelinebetweentwopointsonthelineistermedalinesegment.Alinesegmentisnamedbythetwoendpoints.Thus,inFig.2-2-2,werefertoABaslinesegmentoflinel.Whennoconfusionmayresult,theexpression“linesegmentAB”isoftenreplacedbysegmentABor,simply,lineAB.
Therearethreekindsoflines:
thestraightline,thebrokenline,andthecurvedline.Acurvedlineor,simply,curveislinenopartofwhichisstraight.Abrokenlineiscomposedofjoined,straightlinesegments,asABCDEofFig.2-2-3.
3.Partsofacircle.Acircleisaclosedcurvelyinginoneplane,allpointsofwhichareequidistantfromafixedpointcalledthecenter(Fig.2-2-4).Thesymbolforacircleis⊙.InFig.2-2-4,Oisthecenterof⊙ABC,orsimplyof⊙O.Alinesegmentdrawnfromthecenterofthecircletoapointonthecircleisaradius(plural,radii)ofthecircle.OA,OB,andOCareradiiof⊙O,Adiameterofacircleisalinesegmentthroughthecenterofthecirclewithendpointsonthecircle.Adiameterisequaltotworadii.Achordisanylinesegmentjoiningtwopointsonthecircle.EDisachordofthecircleinFig.2-2-4.
Fromthisdefinitionisshouldbeapparentthatadiameterisachord.Anypartofacircleisanarc,suchasarcAE.PointsAandEdividethecircleintominorarcAEandmajorarcABE.Adiameterdividesacircleintotwoarcstermedsemicircles,suchasAB.Thecircumferenceisthelengthofacircle.
SUPPLEMENTARuler-and-compassconstructions
Anumberofancientproblemsingeometryinvolvetheconstructionoflengthsoranglesusingonlyanidealisedrulerandcompass.Therulerisindeedastraightedge,andmaynotbemarked;thecompassmayonlybesettoalreadyconstructeddistances,andusedtodescribecirculararcs.
Somefamousruler-and-compassproblemshavebeenprovedimpossible,inseveralcasesbytheresultsofGaloistheory.Inspiteoftheseimpossibilityproofs,somemathematicalamateurspersistintryingtosolvetheseproblems.Manyofthemfailtounderstandthatmanyoftheseproblemsaretriviallysolubleprovidedthatothergeometrictransformationsareallowed:
forexample,squaringthecircleispossibleusinggeometricconstructions,butnotpossibleusingrulerandcompassesalone.MathematicianUnderwoodDudleyhasmadeasidelineofcollectingfalseruler-and-compassproofs,aswellasotherworkbymathematicalcranks,andhascollectedthemintoseveralbooks.
SquaringthecircleThemostfamousoftheseproblems,“squaringthecircle”,involvesconstructingasquarewiththesameareaasagivencircleusingonlyrulerandcompasses.Squaringthecirclehasbeenprovedimpossible,asitinvolvesgeneratingatranscendentalratio,namely1:
√π.
Onlyalgebraicratioscanbeconstructedwithrulerandcompassesalone.Thephrase“squaringthecircle”isoftenusedtomean“doingtheimpossible”forthisreason.Withouttheconstraintofrequiringsolutionbyrulerandcompassesalone,theproblemiseasilysolublebyawidevarietyofgeometricandalgebraicmeans,andhasbeensolvedmanytimesinantiquity.
DoublingthecubeUsingonlyrulerandcompasses,constructthesideofacubethathastwicethevolumeofacubewithagivenside.Thisisimpossiblebecausethecuberootof2,thoughalgebraic,cannotbecomputedfromintegersbyaddition,subtraction,multiplication,division,andtakingsquareroots.
AngletrisectionUsingonlyrulerandcompasses,constructananglethatisone-thirdofagivenarbitraryangle.Thisrequirestakingthecuberootofanarbitrarycomplexnumberwithabsolutevalue1andislikewiseimpossible.
Constructingwithonlyruleroronlycompass
Itispossible,asshownbyGeorgMohr,toconstructanythingwithjustacompassthatcanbeconstructedwithrulerandcompass.Itisimpossibletotakeasquarerootwithjustaruler,sosomethingscannotbeconstructedwitharulerthatcanbeconstructedwithacompass;butgivenacircleanditscenter,theycanbeconstructed.
ProblemHowcanyoudeterminethemidpointofanygivenlinesegmentwithonlycompass?
SUPPLEMENTBArchimedesandOntheSphereandtheCylinder
Archimedes(287BC-212BC)wasanAncientmathematician,astronomer,philosopher,physicistandengineerbornintheGreekseaportcolonyofSyracuse.Heisconsideredbysomemathhistorianstobeoneofhistory'sgreatestmathematicians,alongwithpossiblyNewton,GaussandEuler.
Hewasanaristocrat,thesonofanastronomer,butlittleisknownofhisearlylifeexceptthathestudiedunderfollowersofEuclidinAlexandria,EgyptbeforereturningtohisnativeSyracuse,thenanindependentGreekcity-state.SeveralofhisbookswerepreservedbytheGreeksandArabsintotheMiddleAges,and,fortunately,theRomanhistorianPlutarchdescribedafewepisodesfromhislife.Inmanyareasofmathematicsaswellasinhydrostaticsandstatics,hisworkandresultswerenotsurpassedforover1500years!
Heapproximatedtheareaofcircles(andthevalueof¼)bysummingtheareasofinscribedandcircumscribedrectangles,andgeneralizedthis"methodofexhaustion,"bytakingsmallerandsmallerrectangularareasandsummingthem,tofindtheareasandevenvolumesofseveralothershapes.ThisanticipatedtheresultsofthecalculusofNewtonandLeibnizbyalmost2000years!
Hefoundtheareaandtangentstothecurvetracedbyapointmovingwithuniformspeedalongastraightlinewhichisrevolvingwithuniformangularspeedaboutafixedpoint.Thiscurve,describedbyr=a
inpolarcoordinates,isnowcalledthe"spiralofArchimedes."Withcalculusitisaneasyproblem;withoutcalculusitisverydifficult.
ThekingofSyracuseonceaskedArchimedestofindawayofdeterminingifoneofhiscrownswaspuregoldwithoutdestroyingthecrownintheprocess.Thecrownweighedthecorrectamountbutthatwasnotaguaranteethatitwaspuregold.ThestoryistoldthatasArchimedesloweredhimselfintoabathhenoticedthatsomeofthewaterwasdisplacedbyhisbodyandflowedovertheedgeofthetub.Thiswasjusttheinsightheneededtorealizethatthecrownshouldnotonlyweightherightamountbutshoulddisplacethesamevolumeasanequalw
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